Polarization preserving front projection screen microstructures

ABSTRACT

Polarization preserving front projection screens and diffusers provide optimum polarization preservation for stereoscopic 3D viewing, as well as improved light control for enhanced brightness, uniformity, and contrast for both 2D and 3D systems. Generally, the disclosed screens direct light from a projector toward viewers within a diffusion locus, while maintaining optimum gain characteristics. More specifically, light incident on a region of the front projection screen from a predetermined projection direction is reflected by an engineered surface to a predetermined diffusion locus after undergoing substantially single reflections. The engineered surface, comprised of generating kernels, is used to optimally diffuse illumination light into a range of viewing angles, within the diffusion locus, with suitable gain profile, while optimally preserving polarization for 3D applications. Such a screen, when combined with matched polarization analyzing eyewear, provides extremely low cross-talk from any observation point.

CROSS-REFERENCE TO RELATED APPLICATIONS

This is a continuation application and claims priority to U.S. patentapplication Ser. No. 12/538,642, filed Aug. 10, 2009 entitled“Polarization preserving front projection screen material” which is adivisional application of U.S. patent application Ser. No. 12/361,532,filed Jan. 28, 2009 entitled “Polarization preserving front projectionscreen,” now U.S. Pat. No. 7,898,734, that is related to and claimspriority to provisional patent application 61/024,138, filed Jan. 28,2008 entitled “Polarization preserving front projection screen” whichare herein incorporated by reference for all purposes.

BACKGROUND

1. Technical Field

This disclosure generally relates to the front-projection screens andmore specifically relates to front projection screens that optimallymanage the diffusion of light such that polarization is preserved. Suchscreens may additionally maximize image brightness and contrast subjectto specific projector and observation angles.

2. Background

In stereoscopic 3D systems utilizing passive polarization analyzingeyewear, the screen is an integral part of the system. Anydepolarization occurring at the screen results in cross-talk, where theimage intended for one eye is partially transmitted to the opposite eye.This cross-talk is manifested as a “ghost image,” which erodes thequality of the experience and creates eye fatigue. As such, it isdesirable to provide extremely low cross-talk under the most extremeillumination and observation angular conditions.

Known front-projection screens, such as those used in the 2D cinema, arevirtual Lambertian scatterers. Owing to the statistics of the surfaceroughness of such known screens, they have very poor polarizationpreservation and poor effective light efficiency (i.e., while totalintegrated scatter, or TIS, is high, utilization of light in angle spaceis poor).

A known technique for providing stereoscopic 3D polarization preservingscreens is to spray-paint aluminum flake in a transparent binder onto aPVC substrate. Such statistical surfaces provide limited control ofscreen gain profile, directionality, and polarization. Moreover, coatingprocesses frequently show resolvable structures (e.g. sparkle), anduniformity problems, such as textures. Such “silver screens” arefrequently delicate, and are not able to withstand a mild abrasivecleaning process.

Lambertian screens provide uniform appearance in observed brightness,but make poor use of projection light. That is, a significant portion ofincident light is scattered outside of the field of view, reducingsystem efficiency. Moreover, a portion of scattered light is directedback to the screen, reducing contrast and color saturation.

Accordingly, there is a need for a front projection screen which isengineered to optimally disperse light into a range of observationangles, such that the input state of polarization is accuratelypreserved.

BRIEF SUMMARY

This disclosure pertains to engineered reflective diffusers, and inparticular to screens used in front projection systems. The screensprovide optimum polarization preservation for stereoscopic 3D viewing,as well as improved light control for enhanced brightness, uniformity,and contrast for both 2D and 3D systems. The present disclosure seeks todirect light where it is desired, while maintaining optimum gaincharacteristics.

According to the present disclosure, an engineered surface is used tooptimally disperse illumination light into a range of viewing angles,within a specific diffusion locus, with suitable gain profile, whileoptimally preserving polarization. Such a screen, when combined withmatched polarization analyzing eyewear, provides extremely lowcross-talk from any observation point.

Disclosed in the present application is a method for providing apolarization preserving reflective diffuser, wherein the diffuserprovides light to a desired diffusion locus, subject to specificillumination conditions in a manner that preserves polarization. Aviewing locus that includes substantially all viewing locations islocated within the diffusion locus.

According to an aspect, the present application discloses a method forproviding a polarization preserving front projection screen, wherein thescreen provides light to a desired viewing range, or a locus ofobservation, in an auditorium, subject to projector illuminationconditions. The method includes determining a locus that accounts forextremes of illumination and observation angle, that can providesubstantially orthogonal polarization states to all viewing locations inthe auditorium. The method also includes providing a plurality ofreflective generating kernels and distributing the plurality ofgenerating kernels over a substrate.

According to another aspect, a set of design rules are used to produce agenerating function for the surface topography. The generating functionprovides the fundamental building blocks of the microstructure(comprising either one or a plurality of generating kernels), whichcarries with it the (macroscopic) ensemble statistics of the desireddiffuser. Such a design can have superior uniformity in appearance,because it is statistically complete at the fundamental dimension. Thedesign rules further provide for substantially only single reflectionswithin a prescribed diffusion locus. The diffusion locus is defined inaccordance with the angular extremes in illumination anddetection/observation. In yet further design rule considerations, thegenerating kernel can provide a specific intensity distribution (e.g.Lambertian) within the diffusion locus, with a prescribed angular decayin intensity at the boundary of the diffusion locus. In a preferredembodiment that maximizes light efficiency, this decay is represented bya step function.

Other aspects will become apparent with reference to the disclosureherein.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A is a schematic diagram illustrating a side view of a typicalmovie theatre, in accordance with the present disclosure;

FIG. 1B is a schematic diagram illustrating a top-down view of a movietheatre, in accordance with the present disclosure;

FIG. 2 is a schematic diagram illustrating the operation of an exemplarythree-dimensional movie projection system, in accordance with thepresent disclosure;

FIG. 3 is a graph illustrating the polarization-preserving performanceof a conventional silver screen as a function of viewing angle;

FIG. 4 is a graph illustrating the polarization-preserving contrastperformance of a conventional silver screen as a function of viewingangle;

FIG. 5 is a graph illustrating the gain curve of a conventional silverscreen as a function of viewing angle;

FIG. 6A is a polar graph illustrating the viewing locus of a specificauditorium defined by tracing the perimeter of the screen from the frontleft seat, in accordance with the present disclosure;

FIG. 6B is a polar graph illustrating the viewing locus of a specificauditorium defined by tracing the perimeter of the screen from the rearleft seat, in accordance with the present disclosure;

FIG. 7 is a polar graph illustrating the diffusion locus for a randomsampling of theatre auditoriums, in accordance with the presentdisclosure;

FIG. 8 is a graph of the contrast ratio for circular polarization orworst-case azimuth of linear polarization as a function of facetincidence angle, in accordance with the present disclosure;

FIG. 9 is a graph illustrating the differences between contrast ratiofor linear and circular polarizations, in accordance with the presentdisclosure;

FIG. 10 is a graph of a concave structure with a uniform probabilitydensity function, in accordance with the present disclosure;

FIG. 11 is a schematic diagram of a periodic structure with a uniformprobability density function, in accordance with the present disclosure;

FIG. 12A-12D are schematic diagrams of a side view of a theater with aprojector, screen, and seating area, in accordance with the presentdisclosure;

FIGS. 13A-13B are graphs of exemplary gain curves for engineered screensin which light is diffused into the diffusion locus, in accordance withthe present disclosure;

FIG. 14 is a polar plot of a facet-normal locus, relative to the screensurface normal, that substantially illuminates the entire viewing regionwith light from the projector, in accordance with the presentdisclosure;

FIG. 15 is a graph illustrating an exemplary Gaussian surface, inaccordance with the present disclosure;

FIG. 16 is a graph illustrating density of rays reflected from anexemplary Gaussian surface, in accordance with the present disclosure;

FIG. 17 is a graph illustrating an intensity plot of rays thatexperience double reflection from an exemplary Gaussian surface, inaccordance with the present disclosure;

FIG. 18 is a graph illustrating contrast against gain for a series ofsimulations with different amplitudes for Gaussian diffuser surfaces, inaccordance with the present disclosure;

FIGS. 19A-19D provides schematic diagrams showing plots of reflectionconditions for different spacings between Gaussian peaks, in accordancewith the present disclosure;

FIG. 20 is a graphic diagram illustrating a computed locus ofseparations for two Gaussian peaks in which no multiple reflectionsoccur, in accordance with the present disclosure;

FIGS. 21A to 21C are schematic diagrams showing, in each case, thesuperposition of two Gaussian peaks with varying heights and widths, inaccordance with the present disclosure;

FIG. 22 is a graph of a simulated noise pattern, in accordance with thepresent disclosure;

FIGS. 23A to 23D are graphs of the gains and contrasts of both adiffuser composed of two patterns and different characteristic sizesversus a diffuser composed of one pattern and without differentcharacteristic sizes, in accordance with the present disclosure;

FIGS. 24A to 24B are graphs illustrating overlapping functions, inaccordance with the present disclosure;

FIGS. 25A to 25C are graphs illustrating an exemplary generating kernel,in accordance with the present disclosure;

FIG. 26 is a graph illustrating the gain calculated for atwo-dimensional Lambertian generating kernel, in accordance with thepresent disclosure;

FIG. 27 is a graph illustrating radially-averaged gain for a generatingkernel, in accordance with the present disclosure;

FIG. 28 is a schematic diagram illustrating an exemplary hexagonallattice configuration, in accordance with the present disclosure;

FIG. 29 is a schematic diagram illustrating the cell overlap of ahexagonal lattice of generating kernels, in accordance with the presentdisclosure;

FIG. 30 is a diagram illustrating the cell overlap of a square lattice,in accordance with the present disclosure;

FIG. 31 is a schematic diagram of a hexagonal lattice with randomizedcenters, in accordance with the present disclosure;

FIG. 32 is a schematic diagram of a hexagonal lattice with smaller cellsdispersed in between the larger cells with lattice point randomization,in accordance with the present disclosure;

FIG. 33 is a schematic diagram of a semi-regular tessellation lattice,in accordance with the present disclosure;

FIG. 34 is a diagram of randomization with horizontal displacement, inaccordance with the present disclosure;

FIG. 35 is a graph of the probability distribution for cell center tocell center displacement for a surface with randomized horizontaldisplacement, in accordance with the present disclosure;

FIG. 36 is a graph of diffusion angle as a function of separation ofGaussian peaks, in accordance with the present disclosure;

FIG. 37 is a graph illustrating diffusion angle cutoff for overlappinggaussian features, in accordance with the present disclosure;

FIGS. 38A and 38B are graphs illustrating engineered Lambertian diffuseroverlap in two configurations, in accordance with the presentdisclosure;

FIGS. 39A and 39B illustrate one method of pre-correcting the generatingkernels to address overlap, in accordance with the present disclosure;

FIG. 40 is a graph of the pre-corrected cell with Lambertian diffuseroverlap, in accordance with the present disclosure; and

FIGS. 41A and 41B are graphs of gain profiles, in accordance with thepresent disclosure.

DETAILED DESCRIPTION

FIG. 1A is a schematic diagram illustrating a side view of a typicalmovie theatre 100 and FIG. 1B is a schematic diagram showing a top-downview of a movie theatre 100. Movie theatre 100 includes a reflectivescreen 110, a projector platform 120, and a viewing area 130. Projectorplatform 120 may include projector 121 and polarization switch 122.Viewing area 130 may provide seats organized in rows away from thescreen, defining a viewing area or viewing for viewers that may sit (orstand) in different places within the viewing area 130. For instance, afirst viewer may be located at the front-left viewing position 132 ofthe movie theatre 100, and receive reflected light 142. A second viewermay be located at the rear-left viewing position 134 and receivereflected light 144. A third viewer be located in a central viewingposition 136.

With recent developments of polarization technology, there has been aresurgence of three dimensional movies, decoded with matched eyewear.Known three dimensional projection systems project left- and right-eyeimages sequentially using orthogonal polarizations. Forthree-dimensional movie applications that use a single projectorplatform 120, a polarization switch 122 may be placed in the light pathfrom the projector 121 after the projection lens. Such polarizationswitches are known, such as commonly-assigned U.S. Pat. No. 4,792,850(expired), entitled “Method and system employing a push-pull liquidcrystal modulator” to L. Lipton et al., and commonly-assigned U.S.patent application Ser. No. 11/424,087, entitled “Achromaticpolarization switches” to M. Robinson, both hereinafter incorporated byreference. In alternative projector platforms, two or more projectorsmay be used, one to provide left-eye imagery with one polarization stateand the other to provide right-eye imagery with the orthogonalpolarization state. Conventional reflective screens include silverscreens that reflect the polarized light from the projector 120 to themoviegoer.

Typical 3D cinema systems are relatively light starved. A fourteenfoot-lamberts of brightness to the audience will typically providesubstantially lower brightness in 3D mode. One reason for this, forexample, is that sequential systems typically suffer both a polarizationloss (usually greater than 50%) and a time-sharing loss (usually greaterthan 50%), so such systems are typically delivering below 25%brightness, or 3.5 foot-lamberts without a gain screen. Recentdevelopments to address the problem, such as the RealD XL system andcommonly-owned U.S. patent application Ser. No. 11/864,198, entitled“Polarization conversion system for cinematic projection,” by M. Schuck,G. Sharp & M. Robinson, filed Sep. 28, 2007 (herein incorporated byreference), provide a polarization recovery function, but there remainsa desire to increase brightness while preserving polarization.

In systems calling for polarization preservation, total integratedscatter (TIS) from a typical silver screen is approximately 40%, furtherreducing efficiency. While the gain of the screen is high (2.2-2.5 onaxis), from a centered viewing position, the overall perceived imagebrightness is impacted due to rapid fall-off with viewing angle.Conversely, matte screens deliver high TIS (>90%), but make poor use oflight in angle space. Generally, disclosed embodiments seek to maximizeimage brightness by exploiting both high total integrated scatter(approximately greater than 85%), as well as diffusion angle control.Such screens can improve the efficiency of both 2D and 3D experiences.

FIG. 2 is a schematic diagram illustrating the operation of an exemplarystereoscopic three-dimensional movie projection system 200 using asingle-projector (sequential) platform 220. In operation, left-eyeimages 202 and right-eye images 204 may be projected sequentially fromthe projector 220 through polarization switch 222 toward apolarization-preserving screen 210. Polarization-preserving screen 210allows the polarized light from the projector 220 and polarizationswitch 222 to be reflected to the moviegoer 240. The left- and right-eyeimages are viewed by the moviegoer 240 wearing eyewear 250 that decodethe respective orthogonally polarized light to create the experience ofdepth for object 206.

Generally, the quality of the stereoscopic viewing experience dependsupon the ability of the screen 210 to preserve the high degree ofpolarization transmitted by the projector platform 220. Typical matte(near-Lambertian) cinema screens are generally not suitable for use withsuch 3D systems because the scattering is largely diffuse. Owing tostatistics of feature size/height and slope probability density relativeto the illuminating wavelength, such screens are almost completelydepolarizing. However, a high quality stereoscopic 3D experiencepreferably uses at least a 100:1, and more preferably, a 200:1 or highercontrast ratio between the transmitted and blocked images, respectively.

To date, in order to preserve polarization, so-called “silver screens”have been used. Silver screens involve spray painting apoly-vinyl-chloride (PVC) substrate, which may or may not have embossedsurface features, with aluminum flake dispersed in a transparent binder.The tendency is for the facets of the flake to lie nearly parallel tothe substrate plane, thus generating a relatively high specularreflectance and gain with a matte substrate. Efforts to softenhot-spots, or over-saturation at certain points of a screen with darkerperipheries, and to reduce gain often result in a tradeoff betweenappearance, uniformity, and cross-talk. For instance, matting agents canbe included which randomize the air/binder interface, thus decreasingthe hot-spot associated with specular reflection. When evaluatingperformance in the direction normal to the screen 110 (i.e., towardviewing position 136 of FIG. 1), it is common to have linearpolarization contrast-ratio in excess of 150:1. But this falls rapidlywith angle, predominantly because of loss in image brightness. As aresult, there are frequently seats in a cinema auditorium for which thecontrast may sometimes fall below 20:1 in certain locations, such asviewing positions 132 and 134.

Other problems associated with the current metal flake screens are the“graininess” and “speckle” arising from the finite sized anduncontrolled statistical arrangement of the flakes. For the conventionalmatte screen, scattering is accomplished via a high density of extremelysmall scattering centers. Consequently, ergodic statistics are achievedover a relatively small spatial region of the diffuser and theappearance is uniformly white. By contrast, a flake screen consists ofmacroscopic (greater than a micron) features and so require aconsiderably larger region to encompass the same ergodic statistics.Typically this region is larger than the resolution of the human eye andso spatial variation in the scattered intensity is readily visible,i.e., the surface appears “grainy.” As the scattered angle increases,the relative number of facets contributing to the intensity decreases,thereby exacerbating the “graininess” and “speckle” problems.

A coherent contribution to the uniformity also becomes apparent in aflake screen. Despite the incoherent nature of the original projectionlight source, after propagation over the length of an auditorium, theillumination achieves a high degree of collimation and thus a relativelylarge transverse spatial coherence (as large as several hundredmicrons). Facets located within this coherence length can interfereconstructively or destructively to modulate the perceived intensity in asubstantially chromatic way. This is observable on a conventional silverscreen as a faintly colored speckle pattern that is superimposed on theoverall graininess of the screen. However, because the interferenceeffect depends very sensitively on the angle of reflection, the specklepattern appears to move relative to the screen when the observer shiftstheir head. The temporal coherence of the light remains small though,and therefore in order to experience an interference effect, thecontributing facets should be located approximately coplanar to theincident and reflected wavefronts, i.e., the effect is maximized in theretro-reflecting direction and decreases as the scattered angleincreases.

Polarization Contrast Ratio and Gain

The contrast associated with cross-talk is given as the ratio ofbrightness observed for light passing through the transmitting lens tothat which passes through the blocking lens. Variables affecting thepolarization contrast ratio (PCR) include polarization basis vector,projection geometry, observation position, and point observed on thescreen. With a Lambertian screen, the term in the numerator is virtuallyconstant with observation position. But with conventional silverscreens, the gain is sufficiently high that the fall-off in thenumerator term often dominates the angular dependence of PCR. One way ofcharacterizing a screen is to measure the polarization sensitivebi-directional reflectance distribution function (BRDF), which is thereflectivity per solid angle.

FIG. 3 is a graph 300 illustrating the polarization-preservingperformance of a conventional silver screen as a function of viewingangle. Graph 300 shows a BRDF measurement of a conventional silverscreen using a collimated source (HeNe laser with a 0.633 μmwavelength), where a P-oriented polarizer is inserted in theillumination path, and either a P- or S-oriented polarizer is used inthe detection path. P and S are unit vectors parallel and perpendicularto the plane of incidence in the global (substrate) coordinate system,respectively. This should not be confused with the local coordinatesystem, which is associated with individual reflecting facets embeddedin the screen. To obtain these measurements, the screen was illuminatedat −5° off-normal (corresponding to −10° on the plot), so that thespecular direction corresponds to 0°. The detector scanned the in-planeangles, where drop-outs occurred due to the finite size of the detectionmodule.

In FIG. 3, the PP plot 302 corresponds to the parallel polarizer BRDF,which closely tracks the gain profile. The PS plot 304 is thecrossed-polarizer BRDF, corresponding to the power converted toS-polarization through the combination of several mechanisms, as afunction of scatter angle. This term is relatively “white” in anglespace, as would be expected for a diffuse scatter component.

FIG. 4 is a graph 400 illustrating the polarization-preserving contrastperformance of a conventional silver screen as a function of viewingangle. The polarization contrast ratio (PCR) 402 is plotted as afunction of observation angle and is a ratio of the PP BRDF plot 302 tothe PS BRDF plot 304 shown in FIG. 3. It will be shown later that thiscorresponds to a “best case” contrast for linear polarization from aFresnel standpoint, as the input polarization is contained in the planeof incidence.

FIG. 5 is a graph 500 illustrating the gain curve of a conventionalsilver screen as a function of viewing angle. The gain curve 502 showsthe ratio of the PP BRDF to that of a Lambertian scatterer, and as suchis independent of polarization. For this screen, the contrast is halvedat about 20°. Because the PCR tracks the gain, high gain screenstypically show the highest spatial non-uniformity in observedcross-talk.

The above measurements shown in FIGS. 3-5 illustrate that withconventional silver screens, that the BRDF is nearly independent ofincidence angle in the specular direction. The numerator of the PCR of again screen thus depends primarily on the angular difference between theobservation ray and the specular direction. The specular directioncorresponds to the direction the ray would travel if the screen surfacewere a mirror.

Factors that determine the cross-talk leakage term (denominator)include:

1. Depolarization due to diffuse scatter from features much smaller thanthe illuminating wavelength. This may include surfaces of reflectingparticles which are nano-scale rough, sharp edges of particles, andvoids in coatings which expose the underlying matte substrate.

2. Polarization change due to local anisotropy of binder or additivematerials.

3. Polarization change on (specular) reflection from a single surface.

4. Multiple reflections which, on an optical scale, result from surfaceswhich are highly sloped with respect to the illumination direction.

The present disclosure seeks to overcome the limitations in contrastassociated with conventional statistical surfaces such as conventionalsilver screens. Engineered surfaces in accordance with the presentdisclosure may provide more desirable gain profiles using all-reflectivedispersion means, which do not exhibit excessive reflectivity in thespecular direction. Contributions due to the mechanisms listed above canbe severely minimized, if not virtually eliminated. Moreover, control ofthe slope probability density function allows each observer to have asimilar high-contrast experience via improved uniformity in brightness.Finally, engineered surfaces may allow enhanced image brightness, bydirecting projection light to seat locations. This further improvescolor saturation and image contrast by reduction in stray light. Usingthe processes described herein, screen material can be manufactured withthe highest quality at the lowest possible price.

Factor 1—depolarization due to diffuse scatter from features muchsmaller than the illuminating wavelength—refers to depolarizationassociated with interaction of incident light with surfaces that areapproximately on the scale of a few nanometers, to a few hundrednanometers. The contribution of this term tends to be virtually white in(projection and observation) angle space and insensitive to polarizationbasis vector. When observed under a crossed-polarizer microscope, thecontribution appears as a background “glow.” This term can be virtuallyeliminated through the use of high quality optical coatings (low rmsroughness) which are substantially free of voids and are conformal to anembossed surface topography which is free of features at this level.

Factor 2—polarization change due to local anisotropy of binder oradditive materials—is associated with optically thick “transparent”coatings. Such coatings can have anisotropy, which modifies the localstate of polarization. The teachings of the present disclosure mayeliminate this contribution by using single-surface reflection from amirror-like metal coating. Any additional layers may be lowbirefringence oxide-like dielectrics which are applied relatively thin,having virtually zero retardation.

Factor 3—polarization change on (specular) reflection from a singlesurface—refers to the geometry of the local reflecting surface and is aresult of fundamental differences in the complex reflection of S and Ppolarizations. The associated loss in PCR is relatively insignificantfor the typical angles between projection/observation for most cinemaenvironments, but can become significant in more demanding situations.It will be shown that additional conformal dielectric coatings over themetal surface can further reduce this contribution.

Factor 4—multiple reflections resulting from surfaces that are highlysloped with respect to the illumination direction—refers to multiplereflections that can occur (and in certain situations are at a maximum)at normal incidence/observation. They are usually associated with highlysloped diffusing structures. That is, a ray that continues in theforward direction after a single reflection, or does not clear adjacentstructures after a single reflection, undergoes a secondary reflection.The mean-free path between such events can be much larger than thereflecting feature size, thus leading to other undesirable (imagequality) effects. A double-reflected ray can have a highly alteredpolarization state, thus degrading polarization contrast ratio.Moreover, the impact of such reflections is a function of thepolarization basis vector, as will be demonstrated.

Diffuse Scatter

Factor 1 can be virtually eliminated by using continuousmicro-reflective structures that contain little or no contribution atthe high spatial frequencies associated with diffuse scatter.Theoretically, this can be partially accomplished using some of thedesign capabilities described by Morris et al. in U.S. Pat. No.7,033,736 (herein incorporated by reference), where arbitrary slopeprobability density functions can be generated, typical of diffusescatterers, using all-reflective means. Further, these structures canhave pseudo-random distributions in size, location, slope, and heightwhich ensure a matte appearance without compromising performance.

From a practical standpoint, the engineered structures (diffusers) ofthe present disclosure should preferably be mass-produced consistentlyin a manufacturing environment. This may involve roll-to-roll embossingof generating kernels consistent with the specifications describedherein. Moreover, subsequent coatings should preferably be applied witha similar high level of quality, for example, by evaporation orsputtering. Although the disclosure describes the use of thediffuser/screens in a cinema environment, it is contemplated that theymay be alternatively be used in other environments where visual media isviewed, such as, but not limited to home theatre, gaming systems,virtual reality, flight simulators, etcetera.

Current statistical surfaces (e.g., conventional silver screens)invariably have reflector fill-factor below the desired 100%, wherefill-factor is defined as the ratio of metalized area to total area.Here, the metalized area is assumed to have zero transmission. But inthe event that the reflector is partially transmitting, anotherdepolarization mechanism can come into play. More typically withsilver-screens, small pinholes in the coating expose the depolarizingmatte substrate (which is often white). In the event that pinholes areunavoidable, it is desirable to use a highly absorbing base substrate(e.g., matte black) since that will lead to a significantly reducedtransmission of depolarized light. Screens that are manufactured using,for example, gravure printing processes usually have low fill-factor, sotheir PCR is dominated by substrate depolarization.

In a practical implementation, diffuse scatter is often the result ofattempts to eliminate hot spots. Because of the tendency for facets tolie parallel to the substrate with statistical surfaces, effort shouldbe made to spoil the reflection in the specular direction. This can bedone by increasing diffuse scatter, but it is at the expense ofbrightness and PCR. According to the present disclosure, the probabilitydensity function is engineered in such a way as to be uniform in thevicinity of the specular direction. This allows polarizationpreservation while at the same time increasing screen brightness.

Theatre Geometry and Polarization Change on Reflection

An important aspect of designing an optimized statistical surface is afull understanding of the range of geometries associated with cinemaauditoriums. In modern projection booths, the projection lens is(nominally) centered horizontally with respect to the screen, but istypically located above-center vertically. This can range from zero tomore than a half-screen offset. It is typical for the screen to havesingle-axis curvature (about the vertical) with a radius of curvatureequal to (optimally) or exceeding the throw distance. This is in fact arequirement for SMPTE compliance when using a screen with gain above1.3.

It is typical to find stadium seating in two sections, with the frontsection sloped at about 8-10°, and the larger rear section sloped atapproximately 20-22°. The front section is typically curved (like thescreen), while the rear section is typically rectangular. There arefrequently additional seats added in the rows nearest the projector,which increases the effective width of the rear section. In a typicalmovie theatre, the average throw-ratio (ratio of throw distance toscreen width) is approximately 1.8.

In terms of definitions, performance may be described for differentobservation positions from the viewpoint of the hypothetical “idealviewer.” The ideal viewer represents the seat location for which peakbrightness of a white frame occurs at the center of the screen (whenusing a gain screen). Other positions of interest include the perimeterseats for which the system should perform satisfactorily. Theseperimeter seats define the diffusion locus, taken together with theother geometrical considerations discussed above.

Of twenty one cinema auditoriums randomly tested, the average verticaloffset angle of an axial ray is approximately eight degrees down. Thevertical offset biases the specular direction downward, which isbeneficial for brightness with a gain screen. When designed properly,this places the ideal viewer in a central position of the seating.Conversely, with zero-offset, the optimum viewing location with a gainscreen is at the projector, which is clearly not practical. Dependingupon the projector offset and the angle associated with a desired idealviewer, a bias in the diffusion angle may be built into the diffuserdesign, according to the present disclosure.

Worst-case viewing angles are associated with the perimeter seats (orfor a subset of seats for which the system should perform adequately).These seats define the viewing locus. Under ideal circumstance forbrightness and contrast, according to the present disclosure, no lightis thrown outside of the diffusion locus. Moreover, optimizedpolarization contrast ratio requires that only single reflections occurwithin the diffusion locus. In the event that multiple reflectionsoccur, they should preferably occur for reflection conditions outside ofthe diffusion locus.

FIGS. 6A and 6B are polar graphs 600 and 650 respectively illustratingexemplary polar plots for the viewing locus of a specific auditorium atdifferent viewing positions. FIG. 6A shows a plot 602 representing theangle of the observation ray (in the global coordinate system), definedby tracing the perimeter of the screen in the front-left seat (e.g.,viewing position 132 of FIG. 1). FIG. 6B represents the correspondingplot 652 for the rear-left seat (e.g., viewing position 134 of FIG. 1).In this case, the latter (652) is contained within the former (602). Ina typical stadium seating arrangement, however, the rear seats definethe portion of the locus corresponding to the bottom of the screen.

FIG. 7 is a polar graph 700 showing the viewing locus, similar to thosedescribed above in FIGS. 6A and 6B, for a random sampling of twenty onetheatre auditoriums. Such data 704 is contained within a perimeter 702,which, for illumination/viewing conditions symmetric about the vertical,defines a diffusion locus that is also symmetric about the vertical. Itis a design objective to limit diffusion substantially within the regiondefined by perimeter 702, to include the viewing locus, which includessubstantially all viewing positions plus an arbitrary margin of safety,e.g., five degrees.

In the following analysis, assume the screen to include a collection ofmicro-reflectors which, though virtually co-planar on a macroscopicscale, are randomly distributed in orientation in accordance with theslope-probability density function. A local coordinate system is definedhere by a projection ray vector, and an observation ray vector. Thisdefines a local plane of incidence, which contains the local facetnormal vector (where a facet model is typically used for illustration,even though the desired surface may have continuous micro-reflectiveproperties). Because polarization is substantially preserved by thescreen, it is reasonable to assume that light deflected by the facet isthe result of a specular reflection. The likelihood that a facet existswith the desired orientation is given by the two-dimensional slopeprobability density function, which is related to the screen gain.

The local plane of incidence also defines the local S and P vectors (orlocal eigenvectors), which obey Fresnel equations for reflection. Inthis case, the functional coating is typically a metal (e.g., aluminum),which has a complex refractive index, and therefore is absorptive.Assuming that the “facet area” is large with respect to the illuminatingwavelength (or more realistically, that the slope is slowly varying onthe scale of a wavelength) it can be considered that light specularlyreflects from the surface, preserving polarization. As such, there isvirtually no depolarization associated with the event, though a changein the state of polarization (SOP) in general occurs due to the distinctcomplex reflection coefficients associated with S and P.

Consider the specific case where incident linearly polarized lightincludes both S and P projections. A phase difference on reflectiontends to induce ellipticity, while a difference in reflectivity tends torotate the orientation. For a linear polarizer-based 3D system at theworst-case azimuth angle (polarization at ±45° to the facet plane ofincidence), or a circular-polarizer based system at any azimuth angle,the contribution of Fresnel reflection to the polarization contrastratio is given by

${PCR} = \frac{\left( {{\sqrt{R_{P}}/2} + {\sqrt{R_{S}}/2}} \right)^{2} - {\sqrt{R_{P}R_{S}}{\sin^{2}\left( {\Gamma/2} \right)}}}{\left( {{\sqrt{R_{P}}/2} - {\sqrt{R_{S}}/2}} \right)^{2} + {\sqrt{R_{P}R_{S}}{\sin^{2}\left( {\Gamma/2} \right)}}}$where √{square root over (R_(P))}e^(−iΓ/2) and √{square root over(R_(S))}e^(iΓ/2) are the complex reflection coefficients associated withP and S polarizations, respectively (neglecting common phase), where Γis the phase shift between the R and P components. To first order, thefirst term in the denominator accounts for contrast loss due toreflectivity difference, while the second term in the denominatoraccounts for contrast loss due to phase retardation.

Fundamentally, the facet incidence angle should be less than 45°, having(e.g., a flat screen) infinite throw distance (with a centeredprojector) and a viewer located at the plane of the screen. Moretypically in theatre auditoriums, the maximum facet incidence anglesassociated with the worst case viewer are below 35°.

FIG. 8 is a graph 800 of the Fresnel PCR for circular polarization (orworst-case azimuth of linear polarization) as a function of facetincidence angle. The contrast 802 is above 1,000:1 for angles below 25°(which accounts for most of the audience), remaining above 270:1 forangles out to 35°. As such, the Fresnel contribution is relatively smallin present cinema environments.

In terms of relative contributions (again to first order), the contrastdue to reflectivity difference alone is 24,000:1 at 35°, while thecontrast associated with retardation alone is 273:1. Thus, the loss incontrast associated with Fresnel is mostly due to phase shift between Sand P. The opportunity exists to more closely match phase of S and P,while increasing overall reflectivity by adding conformal transparentdielectric layers (that at the same time serve to prevent the growth ofnative oxide) over the metal. This is typically done with so-called“protected aluminum” mirror coatings. Lippey et al., in U.S. Pat. No.7,110,175 (herein incorporated by reference) discloses the deposition ofan aluminum layer to address the reflectivity difference by using adielectric layer to make the reflectivity of S— and P— the same.However, Lippy fails to recognize that contrast is impacted far more byphase difference than the reflectivity difference. Becausedepolarization is assumed to be only a consequence of difference betweenS and P reflectivity, the natural objective for contrast is to minimizethe local incidence angles, and thus to have higher gain. However, asdisclosed herein, we have demonstrated that the dominant depolarizationmechanism is multiple scattering events, which Lippey does not evenmention. In other words, Lippey does not recognize the dominantmechanism that contributes to contrast performance or the techniques forsuch optimization of contrast performance. Furthermore, the secondobjective identified by Lippey is to match the amplitude reflectivity ofS and P polarizations. However, with the teachings of the presentdisclosure, it is possible to both match the amplitude of S and Preflectivities and maximize the phase difference between the twocomponents, thereby providing superior performance in polarizationpreservation properties.

Frequently, dielectric overcoats are deposited onto metal mirrors toprovide durability and enhance reflectivity. If bare aluminum is notcoated, it is easily scratched and will eventually form a thin layer(70-90 Å) of native oxide (Al₂O₃). Native oxide (index n=1.66) will tendto reduce reflectivity over time. If instead, a layer which isapproximately a quarter-wave in optical thickness of MgF, (n=1.38) isdeposited onto the bare aluminum, reflectivity can be made to increaseby a few percent. While MgF₂ represents an ideal choice of dielectricovercoat, substantial improvement can be obtained using other low indexdielectrics such as SiO₂. In both cases, the primary constraint is tominimize the complex phase difference between the S and Preflectivities.

In the present disclosure, polarization preservation is likely moreimportant than increasing efficiency. As discussed above, reducing thephase difference between S and P components through thin-filmcompensation helps preserve polarization. A dielectric film of arbitrarythickness and index, with the performance metric being the PCR at 589 nm(where the input polarization is at 45° to the plane of incidence),produces the best results when the refractive index is minimized (e.g.,with MgF₂). When a native oxide of 70 Å thickness is used, the PCR at afacet incidence angle of 35° is 139:1 (which is lower than barealuminum). Adding a 0.34 wave thickness layer of MgF₂, yields a contrastof 23,915:1. At 28° the contrast is lower for the compensated case(1,934:1), but still, the contrast remains significantly higher than theuncompensated case (360:1). For smaller angles, contrast generallyincreases, but the compensated case remains at least a factor of threelarger than the uncompensated case.

Because a tilted facet has linear eigenpolarizations, performance of asystem based on linear polarization is azimuth dependent. If the inputpolarization is contained in the facet plane of incidence, thenpolarization is preserved on reflection. If this mechanism is importantin determining contrast, then eyewear can be selected to optimizeoverall performance. For instance, screen corners tend to correspond tothe largest facet incidence angles, which may tend to be closer to the±45° azimuth angle than the 0/90° azimuth. In these circumstances, asystem based on ±45° linear polarization eyewear may be used. As forsystems using a circular basis, there is no relief from polarizationchange on reflection for any azimuth because, in fact, the contrast isindependent of azimuth angle. In the event that contrast is dominated bymultiple reflections, then the above argument may not be a relevantdesign consideration.

Double Reflections

Depending upon the screen structure, polarization change after a singlereflection may not be the most important factor influencing cross-talk.Highly directional diffusers, such as those manufactured by WavefrontTechnologies, have highly sloped ridges which tend to produce secondaryreflections. Under the crossed-polarizing microscope, off-the-shelfholographic diffusers typically exhibit linear eigenpolarizations due toretroreflections when illuminated and detected normally. In a test ofthree samples of products normally used in transmission, but coated withaluminum for the test, all samples dispersed substantially more in onedimension than the orthogonal direction (8°/21°, 10°/68°, 12°/44°. Thecoated samples verified that contrast was several hundred to one whenthe input polarization was parallel to the structure axis, but wassubstantially lower when the sample was rotated, with contrasts of onlytens to one in the 45° azimuth. Note that these measurements were madein a retroreflecting arrangement.

Statistical surfaces, such as metal-flake screens, are also prone todouble-reflections. Often, the mean-free path between pairs of facets issubstantially larger than the actual reflecting feature sizes. When ascreen sample is rotated under a crossed linear polarizer microscope ina retroreflecting arrangement, the brightness of facet pairs can beobserved to change in unison. This is likely due topolarization-converted light emerging from opposing propagationdirections. Along the eigendirections, the pairs are highlyextinguished. Due to the high degree of polarization conversion withretroreflection, the pairs become very bright in the ±45° azimuth. Theeffective geometry of the pairs is often very similar (dictated by theoverlap area of the facets), which is another factor that makes themeasily identifiable.

Retroreflecting arrangements have potential benefits from a brightnessstandpoint. That is, if the direction of peak diffusion is, in general,counter to the incident direction, then light from the projector willhave a greater tendency to be thrown toward the audience. Beadedscreens, for example, can have the benefit of functioning like acat's-eye retroreflector. Because they have a self-correction property,retroreflectors can virtually eliminate the need for local control ofdiffusion properties as a means of optimally dispersing the light.However, care should be taken to ensure that such retroreflections donot compromise polarization, as would occur with certain (e.g.,corner-cube) retroreflectors. In the case of double reflections fromfacet-pairs, polarization is substantially converted to the orthogonalstate.

In the past, lenticular-like periodic structures have been used oncinema screens to disperse more in the horizontal direction relative tothe vertical. In the event that structured surfaces are used to disperselight to a greater degree in the horizontal than the vertical, which areprone to secondary reflections, a system based on 0/90° polarizationeyewear may be used. More likely, however, screen structures that areprone to secondary reflections will not perform adequately.

Contribution of Each Term to PCR

Specific measurements can be used to extract the contribution of eachphysical mechanism to the PCR. Based on the above discussion, thediffuse scatter term is likely to be white in angle space, andindependent of polarization basis vector. This is the background leakageterm. As such, PCR results versus facet incidence angle should be thesame for both linear and circular polarizations. If not, then anotherphysical mechanism is likely to be contributing significantly.

The Fresnel contribution is zero in the retroreflecting direction (forsingle reflections), becoming significant as the facet incidence angleapproaches 20°, and growing as angle increases. It is clearlypolarization dependent, vanishing when the input isparallel/perpendicular to the facet plane of incidence, and maximum at±45°. The contribution to PCR is independent of azimuth when usingcircular polarization. Thus, if linear PCR results are a strong functionof azimuth (or if there is a significant difference between linear andcircular PCR) at large incidence angles, then the Fresnel term may beimportant. This assumes that the contribution from multiple reflectionsbecomes relatively insignificant at such large angles (or is separable).

In the event that linear and circular basis vectors give different PCRin the retroreflecting direction, the reason is likely doublereflection. With a statistical surface, where the slope probabilitydensity is uniform in azimuth, the likelihood of a double reflectionevent is likewise uniform in azimuth. For circular polarization, thecontribution to PCR is therefore also uniform in azimuth. But because ofthe azimuth dependence of linear polarization, the contribution to PCRaveraged over the entire azimuth is half that of the circular case.

As a way to test this contribution, polarization sensitive BRDFmeasurements were made for both linear and circular cases, where thelinear case is along an Eigen-direction.

FIG. 9 is a graph showing that the contrast of the linear case 902 isover 160:1, where the circular case 904 is only 110:1. Because thisdifference is observed along the retroreflecting direction, anydifference in PCR may be attributed to multiple reflections, unless thePCR of the circular case is inherently lower. To obtain these results, a532 nm laser was directed through a pair of orthogonal linear andcircular polarizers to test the baseline performance at normalincidence. The baseline PCR was measured to be 888:1 for the linear caseand 895:1 for the circular case, the difference of which is well withinexperimental error and limited by the polarizer. Thus, substantiallyhigher PCR may be provided by the engineered surface of the presentdisclosure through the elimination of double reflections.

Another objectionable aspect of statistical surfaces is the lack ofspatial control of BRDF characteristics. Spatial variation in the facetprobability density function can produce a non-uniform appearance. Thefeature size associated with such variations may be highly dependentupon the manufacturing processes and the statistical control aspects ofeach. Given that the size of a pixel on an average screen for a full-HD(1024×2048 pixel) 2 k projector is roughly 7 mm, significant variationin reflected intensity over this (or larger) sizes is likelyproblematic. To demonstrate this, a screen sample was illuminatednormally at a distance of 305 mm, with an amplitude stable 532 nm laser.The laser and detection module were mounted with a separation of 45 mmon the same rail carrier (in-plane), and were translated in 1 mmincrements along the screen axis. The detector aperture is 5 mm,virtually eliminating the contribution of speckle. Some smoothing of theprofile is assumed to occur as a consequence of averaging over theassociated detection solid angle.

The total scan range in a particular position was 100 mm, with nosignificant difference in results for other positions on the screen.Screen samples from two vendors were tested. For Samples A and B, thestandard deviation in reflected power was 6.8% and 5.2% respectively.The maximum deviation was +21% and −16% for Sample A, and +11% and −14%for Sample B.

An estimate of variation in perceived brightness of pixels, due tonon-uniform BRDF, is obtained by comparing the average power collectedin 7 mm segments of the scan. For Sample A, the average deviation was4.6%, while for Sample B the average deviation was 6.7%. Thecorresponding maximum deviation was 8% and 7.5%, respectively. A benefitof the engineered surface according to the present disclosure is thatsuch fixed-pattern variations could be virtually eliminated at allrelevant scales.

Interference Effects

After propagation from the projector to the screen, and assuming spatialcoherence of projection light on the order of a wavelength, theillumination light can exhibit spatial coherence over areas that aresignificant relative to the screen resolution area. This can exacerbatescreen appearance uniformity problems via coherent superposition on theretina. According to the present disclosure, the engineeredmicrostructrure can have a spatial frequency noise structuresuperimposed on the desired topography. Such a structure does not impactthe single-reflection requirement, but randomizes the phase in such away that light collected by the eye contains a substantially uniformrepresentation in phase space. If the amplitude of the noise is somemultiple of a wavelength and the wavelength is similar to the spatialcoherence length of the light, then the phase randomization should besufficient to substantially reduce speckle.

Matte Appearance

The eye resolves approximately one arc minute below which the perceivedintensity can be considered a weighted integration of the probabilitydensity that generating kernels scatter from the illumination directioninto the observation direction over the associated area. Thisprobability is related to the local slope probability density. In theevent that the integrated probability varies spatially, meaning that thesampled area does not conform to ensemble statistics, the screen willhave a grainy texture, which is objectionable. This can happen both as aconsequence of large feature sizes, as well as their specificdistribution over the screen surface. Often, the relative intensityfluctuations increase with observation angle, where the probability thatthe required slope area exists is substantially diminished relative tothe specular direction.

In conventional cinema screens, a matte appearance is the result of verysmall features, which produce multiple scattering events that contributeto an averaging in angle space. According to the present disclosure, theengineered diffuser is analyzed and modified spatially to create moreuniform intensity distribution after single reflections. Thissubstantially reduces the graininess of the appearance while preservingthe polarization. To a large extent, this is accomplished by theengineered diffuser shape in accordance with the present disclosure.Locally, each scattering feature fills the entire viewing locusergodically. Spatial fluctuations are due mainly to randomization andtiling effects. The percentage of area subject to such effects can besmall relative to the unaffected region, i.e., it can be limited toregions where the engineered generating kernels are overlapping. Byusing generating kernels with zero slope and zero height at theboundary, this effect is substantially limited to the specular region ofthe gain profile where it is much less objectionable.

Exemplary Screen Designs

In the event that the screen contains a statistically homogeneousdistribution of generating kernels, it is necessary that such kernelssubstantially satisfy the extreme conditions of illumination andobservation. Every point on the screen accepts illumination from one (ormore) discrete angles. Over the areas associated with ensemblestatistics, this illumination can typically be considered collimated.For each such illumination area, light should be scattered into a rangeof angles associated with the diffusion locus, subject to gainrequirements. For each such point, an important quantity is the extremeangle formed between the specular direction and the observationdirection. When the scatter requirements for each point of the screenare overlaid, the perimeter, termed here as the “diffusion locus”defines the screen microstructure diffusion requirements. The diffusionlocus is related to the slope probability density of the screenmicrostructure.

When describing a microstructured diffuser, it is convenient to considerthe smallest fundamental structural unit (or units) that is replicatedto form a macrostructure. This structure is referred to herein as thegenerating kernel, and for a surface diffuser it will have sometopographical shape that determines the diffusion profile of lightreflected off of it. In an ideal situation, this generating kernelcarries the entire ensemble statistics of the required diffuser so thatpoint-to-point variation in the diffusion are minimized at the smallestpossible scale. In a more general case, the generating kernel may notfully satisfy these statistics, but an ensemble of such structures may.

An aspect of the present disclosure is to design the profile of agenerating kernel (or microstructure comprising a plurality ofgenerating kernels) to eliminate secondary reflections within the facetincidence angles associated with the full range ofillumination/observation angles. In one embodiment, this is accomplishedby determining the diffusion locus of illumination/observation toprovide light to all required seats in the auditorium (based on thegeometrical considerations discussed previously) and designing agenerating kernel (or microstructure comprising a plurality ofgenerating kernels) that achieves at least one of the following: (1) aslope probability density function that is uniform throughout thediffusion locus (virtually Lambertian), with little, if not no, “spike”in the specular direction; (2) a slope probability density function thatis uniform spatially (e.g., ±1%) so that there is little, if not no,perceived modulation in brightness; (3) a slope probability densityfunction that has a sharp cutoff in angle space at the perimeter of thediffusion locus; (4) a generating kernel layout that is free of featuressmaller than a few microns, such smoothness ensuring that the degree ofpolarization is preserved; (5) generating kernel feature sizes that aresmaller than several hundred microns (which could result in e.g.“grainy” or “sparkle” appearance); and (6) rays incident within theperimeter of the diffusion locus do not undergo any substantialsecondary reflections before entering the diffusion locus.

By providing a sharp cutoff in the slope probability density, it ispossible to eliminate rays that tend to scatter in the direction ofincident light (forward scatter), or adjacent structures. Such lightwill undergo two or more reflections, generally with significant changein polarization. Additionally, light that would otherwise not enter thediffusion locus can be used to increase image brightness, and eliminateloss in color saturation and contrast resulting from stray lightscattered from auditorium surfaces.

FIG. 10 is a graph 1000 of a one-dimensional example of a concavestructure 1002 which has a uniform probability density function with ahard cutoff at 80°. Mathematically, the requirements for such astructure are as follows. First, the rate of change of θ is proportionalto the inverse of the desired scattering probability,

(θ): ∂θ/∂x=c1/

(θ) (where c has units of inverse distance and sets the scale for thegenerating kernels). Second, the hard cutoff in the probabilitydistribution is determined by setting the limits of integration of θ.Finally, the slope at any point on the surface is equal to the tangentof half the scattering angle: ∂z/∂x=Tan(θ/2).

FIG. 11 is a diagram illustrating a 1D structure 1100 which is bothperiodic and satisfies the same criteria. The convex elements 1102 inthe structure are obtained by rotating the concave elements by 180°.Adjacent cells in the structure may have arbitrary size as long as theaspect ratio is preserved, the size remains small enough to be visiblyirresolvable but large enough to prevent diffuse scatter (e.g. less thanseveral hundred microns and greater than a couple of microns), and nomultiple reflections occur within the diffusion locus. Light incident atangles up to 10° suffer no multiple reflections as shown by reflectedray 1104, which clears the adjacent peak. A randomized surface may begenerated by tiling multiple unit cells with different widths as shownin 1106.

More generally, a statistical surface can be generated by eliminatingthe differential equation, ∂θ/∂x=c1/

(θ), but maintaining the slope probability density,

(θ). Reflecting structures may have different shapes as long as thedensity of surface elements with slope θ is equal to

(θ). In particular, this allows the design to accommodate differentscattering requirements in different regions of the screen.

FIGS. 12A-12D are schematic diagrams showing a side view of an arbitrarytheater including the projector, screen, and seating area.

In FIG. 12A, in operation, ray 1204 travels from the projector 1202 tothe bottom part of the screen 1206. In order to illuminate the seatingarea 1208, it should be scattered into the diffusion locus 1210. Thediffusion locus is defined in accordance with the angular extremes inillumination and detection/observation. Within the diffusion locus,substantially only single reflections occur from the screen toward thediffusion locus; whereas outside the diffusion locus, multiplereflections may occur.

FIG. 12B is a schematic diagram showing an example microstructure 1222at the screen surface in FIG. 12A. Rays 1224, 1226, and 1232 are allapproximately parallel to ray 1204 but illuminate different parts of themicrostructure. Rays 1224 and 1226 experience single specularreflections 1230 and 1228 before entering the diffusion locus 1210. Ray1232 experiences two specular reflections but the exiting ray 1234 doesnot enter the diffusion locus 1210 and so will not likely cause adecrease in PCR.

In contrast, FIG. 12C shows a ray traveling up from the projector to thetop of the screen 1206 which illuminates a substantially differentviewing location within the diffusion locus. FIG. 12D illustratesreflections 1268 from a microstructure 1262 located at the top of screen1206 showing that no rays incident on the top part of the screenexperience multiple reflections although some reflected rays 1270 do notenter the diffusion locus 1240.

Thus, compared to rays traveling up from the projector 1202 to the topof the screen 1206, rays traveling down from the projector 1202 andimpinging on the bottom part of the screen 1206 should preferablyscatter into a substantially different portion of the diffusion locus inorder to illuminate the seating area, i.e., the slope probabilitydensity is also a function of incident angle. Furthermore, becausedifferent incident angles illuminate different viewing locations, somedouble reflections may be tolerated as long as they result in light thatdoes not enter the diffusion locus. These effects increase as the throwof the projector is decreased. Although a single microstructure 1222,1262 is shown, consistent with the present disclosure, a singlemicrostructure may be comprised of one or more generating kernels.

FIGS. 13A and 13B are graphs 1300, 1350 of several possible “gain”curves for engineered screens in which light is only scattered into thediffusion locus shown in FIG. 7. Here, the gain is calculated within thediffusion locus and is assumed to be symmetric about the vertical axis,but with a sharp cutoff, as shown in FIG. 7. Graph 1300 illustrates thata uniform (Lambertian-like) profile 1310 within the diffusion locuswould result in an efficiency increase of almost 30% over a typicalmatte white screen profile 1320. Graph 1350 illustrates that if the gainprofile has the same functional shape as the existing gain silverscreen, the increase in efficiency 1360 is almost 100%. Alternatively,the gain curve can be flattened, shown by line 1340, such that theoverall uniformity is better than the conventional silver screen, shownby line 1330, i.e., widen the gain profile of the current silver screen,with substantially the same maximum brightness.

FIG. 14 is a polar plot 1400 of the facet-normal locus, relative to thescreen surface normal, that substantially illuminates the entire viewingregion with light from the projector. At each point on the screen, thereis a set of facet normals 1410 that direct light from the projector toeach individual seat. The union of all such sets across the surface ofthe screen defines the locus of facet normals 1420 to ensure that eachviewer receives light from substantially all parts of the screen. Anyfacet normals falling outside this locus result in wasted light. Theblack dots 1410 are computed from the left side extreme-viewing-locationseats from a random selection of theaters. The locus or curve 1420 isextended to encompass the viewing angles for the right-most seats aswell.

The desired geometrical prbperties for a polarization preservingprojection screen have been identified as: (1) Filling the diffusionlocus with uniform light intensity; (2) Preventing multiple reflectionsof light by (a) introducing a cutoff angle in the light distribution toprevent reflected light from hitting the screen a second time and (b)keeping features with steep slopes well separated so that scatteredlight at large angles does not encounter a second surface; (3) Achieveergodicity within a region smaller than a pixel. i.e., the fulldiffusion locus should be sampled uniformly by an area of the screenthat is smaller than a pixel so that the screen intensity is spatiallyuniform; (4) Ensure that all features are significantly larger than anoptical wavelength to prevent scatter; and (5) Avoid periodic structuresthat could combine with the pixilation of the projector to producemoiré, or interference between sets of fine pattern grids. A curve wasfound that satisfied these requirements for 1D scattering.

As used herein, “ergodicity” is the condition wherein the average valueof some parameter over a finite region has converged to the ensembleaverage of the entire region. When a region of a certain size is said tobe ergodic, it is not statistically different from any like-sized orlarger region in the ensemble.

There are two general strategies to realize a 2D diffuser surface. Thefirst is to determine a set of rules that can be used for random(stochastic) processes that on average satisfies the requirements.Identifying fully random processes that will in general satisfy all ofthe design requirements may be complicated, but the manufacturing ofthese surfaces may, in general, be easier. The second strategy is todesign a custom structure that explicitly satisfies all of the aboverequirements. This ensures the best performance, but requires amanufacturing technique that can transfer this design to the screensurface with high fidelity.

Stochastic Design

There are a multitude of techniques available with which to make asurface diffuser from random structures. These include holographicrecording of laser speckle, chemical etching, mechanical etching (e.g.,bead blasting), and coating with metal flake encapsulated in polymerbinder. The local geometry of the individual scattering features inthese diffusers is defined by the process in which the diffuser iscreated. For example, a holographic diffuser will be composed of 2DGaussian peaks whereas a metal flake paint will be composed of acollection of planar facets with sharp edges. Neglecting any sharpedges, the limit of a large number of such features is expected toconform to Gaussian statistics. Therefore, a stochastic diffuser may beapproximated as a randomly distributed collection of Gaussian scatteringfeatures wherein the features have some characteristic height d, andwidth σ.

In principle, the mean values of d and σ can often be independentlycontrolled. For example, for a laser speckle pattern, σ is thecharacteristic speckle size and can be adjusted by modifying thedistance to the aperture or the size of the aperture. If the specklepattern is recorded in photo-resist, then d can be controlled bymodifying the exposure time and/or the developing conditions. Similarly,in a bead-blasting process, σ will be related to the size of theablating particles and d will be proportional to their incoming velocity(to first-order). Therefore, constructing design rules for thegeneration of stochastic diffusers depends on an understanding of therelationship between d, σ, and the preservation of polarization, i.e.,double reflections.

FIG. 15 is a schematic diagram illustrating a top-down view 1500 of anexemplary Gaussian surface and respective side views 1502 and 1504 thathave been simulated with Gaussian statistics to verify the computationalmodel against experiments on physical samples. A 2D diffuser withGaussian statistics is simulated by populating a plane with randomlypositioned Gaussians (e.g., Gaussians 1506 and 1508). To first order, itis sufficient to use identical Gaussians (σ, d are constants).Relatively uniform coverage is provided by locating the peaks on ahexagonal lattice and then randomly translating their positions by aGaussian weighted distance. If the standard deviation of the translationis sufficiently large then the underlying hexagonal order is erased andthe pair-pair correlations become Gaussian. This results in a Gaussiannoise distribution as shown in FIG. 15. In this exemplary simulation,σ=30 μm, d=16 μm, the underlying lattice constant is 60 μm and the totalwidth of the structure is 2 mm.

The scattering distribution and gain of this structure 1500 wassimulated using non-sequential ray tracing (ASAP) software. Theillumination was a uniform collimated light source that fully sampledthe surface at normal incidence. To speed computation and simplify theanalysis, the polarization of individual rays was neglected andnon-geometrical effects were ignored (e.g., Fresnel reflectivity,scattering from sub-wavelength features). To compute the gain, all ofthe rays that reflected from the surface only once were collected.

FIG. 16 provides a graph 1600 illustrating the density of the raysreflected from the exemplary Gaussian surface of FIG. 15, plotted as afunction of angle. Graph 1600 depicts this 2D ray trace as a Gaussiannoise surface and graphs 1602 and 1604 illustrate the profile fromhorizontal and vertical aspects respectively. In graph 1600, thesimulated rays have only reflected from the surface once and theintensity has been scaled by cos(θ) to show the gain. A gain plot of allof the rays that reflected from the surface twice can be used to computethe depolarization effect of multiple reflections.

FIG. 17 provides a graph 1700 illustrating an intensity plot of raysthat experienced a double reflection from the exemplary Gaussian surfaceof FIG. 15. Graph 1700 and side views 1702, 1704 reveal that theGaussian surface is not fully ergodic, i.e., because the scattereddistribution is not radially uniform this surface is not statisticallysmooth. This is consistent with experiments on holographic diffuserswith similar feature sizes in which significant intensity variation isseen between adjacent 3 mm×3 mm patches. However, a radial average ofthe distribution is a good approximation of the full distribution. Thepolarization contrast ratio due to multiple reflections is the ratio ofthe intensity due to single reflections, as shown in FIG. 16, to theintensity due to double reflections, as shown in FIG. 17.

FIG. 18 is a graph 1800 illustrating contrast against gain for a seriesof simulations with different amplitudes for Gaussian diffuser surfaces.By increasing the amplitude of the noise distribution, the gain of thestructure is decreased. This increases the likelihood of multiplereflections and, as a result, the contrast decreases. The trend isqualitatively similar to trend measured on a series of holographicdiffusers. The line 1802 showing the simulated result has consistentlyhigher contrast than the line 1804 showing the experimental result dueto the lack of point scattering, Fresnel effects, and finitepolarization sensitivity of the measurement system. This series ofexperiments highlights some of the limitations of a statistical surfaceas a diffuser for a cinema screen. There is an intrinsic tradeoff insuch structures between gain (and thus, illumination uniformity) andcontrast. To the extent that high gain can be tolerated, higher contrastcan be obtained. However, it should be noted that to achieve both highcontrast and low gain requires a carefully engineered surface.

In order to diagnose these results, the scattering properties of arandom surface may be calculated. Consider a Gaussian peak with height dand width σ.

$\begin{matrix}{{z(r)} = {{\mathbb{d}{\mathbb{e}}}\frac{r^{2}}{\sigma^{2}}}} & \left. 1 \right)\end{matrix}$The maximum slope on this feature occurs at

$r = \frac{\sigma}{\sqrt{2}}$and gives rise to a reflection angle θ of

$\begin{matrix}{\theta = {2{\tan^{- 1}\left( {\frac{\sqrt{2}}{\sigma}{\mathbb{d}\;{\mathbb{e}}^{- \frac{1}{2}}}} \right)}}} & \left. 2 \right)\end{matrix}$Therefore for a given feature height, a minimum feature width σ_(m) canbe set, such that for isolated scattering features, produces a cutoffangle θ_(c).

FIGS. 19A-19D provide schematic diagrams showing plots of reflectionconditions for different spacings between Gaussian peaks. FIG. 19A is aschematic diagram showing a reflection of a ray 1904 off a singlescattering feature 1902. For a single scattering feature, as long asθ_(c) is less than 90°, the reflected ray 1906 will be directed awayfrom the surface and there will be no secondary reflection. FIG. 19B isa schematic diagram showing the reflective properties as adjacentfeatures 1922 and 1924 get closer. However, as adjacent features 1922and 1924 approach, there is some region in which for large reflectionangles, a second reflection does occur. FIG. 19C is a schematic diagramshowing the situation as the peaks 1932 and 1934 approach even more.Here, the double reflection disappears because the superposition of thetwo peaks decreases the maximum slope in the region between them. FIG.19D is a schematic diagram showing the scenario of the two peaks 1942and 1944 overlapping, such that the maximum slope is increased, which inmany cases leads to a multiple reflection. Therefore, for adjacentGaussian peaks, there exists a locus of regions in which multiplereflections occur.

FIG. 20 is a graph 2000 illustrating that for two Gaussian peaks withequal height and cutoff angle θ_(c), there is a computed locus ofseparations in which no multiple reflections occur. In order for thereto be zero (or close to zero) probability of a double reflection on thesurface defined by the two peaks, θ_(c) of the Gaussian is preferablyless than 52°. For θ_(c) less than 80°, no multiple reflections occurbetween the peaks, but this may occur as the peaks begin to overlap.Should the possibility of three peaks overlapping be considered, thenθ_(c) would be smaller still. Unfortunately, θ_(c) is considerablysmaller than the desired diffusion locus for a typical theater. Lightwill be scattered to larger angles as the peaks approach each other, butto use this behavior to fill the diffusion locus, one would need a highdensity of scattering features. In this case, the probability of morethan two features overlapping increases drastically. In sum, an attemptto eliminate double reflections, by decreasing the θ_(c) to less than52°, results in light not hitting the entire diffusion locus.

One way to solve the problem above is to use Gaussian peaks withdifferent heights and widths. FIGS. 21A to 21C are schematic diagramsshowing the superposition of two Gaussian peaks with d₁=1, σ₁=1 andd₂=⅕, σ₂=⅕. The reflected angle as a function of position is calculatedfor different peak-to-peak separations. When the peaks arewell-separated (see, e.g. FIG. 21A), the reflection angle on the surfaceis locally that of the individual Gaussians (θ_(c)=50). However, as thepeaks approach, the maximum slope between them is enhanced. FIG. 21Bshows the smaller peak is located approximately on the shoulder of thelarger peak. The maximum slope then decreases as the peaks exactlyoverlap (see, e.g., FIG. 21C). The overall effect is similar to thesuperposition of equal sized features with one important difference: thesame maximum slope condition is obtained but the average depth of thesurface has not significantly increased. Consequently the large-anglescatters remain relatively well separated and the likelihood of a secondreflection is smaller.

FIG. 22 is a graph of a simulated noise pattern 2200 illustrating twopatterns composed of structures with different height and width butsubstantially identical cutoff angle.

FIGS. 23A to 23D are graphs showing a comparison of the gains andcontrasts of both a diffuser composed of two patterns and differentcharacteristic sizes versus a diffuser composed of one pattern andwithout different characteristic sizes. The gain calculated from thediffuser with different characteristic sizes yields a much smoothercurve 2302 (see FIG. 23A) than the gain curve 2352 calculated from onlya single periodic structure (see FIG. 23C). This is because the smallerfeature size of the first structure allows the curve 2302 to becomenearly ergodic in a smaller area, but the gain of the two structures isapproximately the same. More importantly, the peak contrast of thetwo-pattern diffuser (see FIG. 23B, showing an enlarged portion of FIG.23A) is substantially five times larger than the peak contrast of thesingle pattern diffuser (see FIG. 23D, showing an enlarged portion ofFIG. 23B). The average contrast is more than twice as large.

In conclusion, a practical technique of increasing the maximum contrastin a purely statistical structure is to superpose two patterns withdifferent periodicities. The practical limitations of this technique arethat the smaller feature should preferably remain large relative to thewavelength of light (e.g., in the order of 10's of μm) and the largerfeature should preferably remain small relative to a pixel (e.g., in theorder of 100's of μm). With holographic diffusers, this can beaccomplished by performing two exposures in which the second exposure isadjusted to have approximately ⅕ of the height and 5 times thefrequency. Another technique to accomplish this would be to apply a highgain metal flake paint to an embossed substrate.

Custom Design

Given the ability to precision enginer the height of the surface on apoint by point basis, a technique for designing that surface is alsodisclosed herein. In principle, it is possible to treat the diffusersurface as a connected web of polygons. One may then perform aMonte-Carlo simulation to find the optimum orientation and height ofthese polygons that optimizes the diffusion characteristics of thesurface. However, since ergodicity is desired over a small region,limitation on minimum feature size, as well as the constraints onmultiple reflections makes this an unnecessarily expensive computation.It is more practical to instead use a specific generating function thatis then replicated over the surface of the diffuser. This function canbe generic, such as a Gaussian, in which case the statistics shouldpreferably be constrained in an essentially non-Gaussian way in order tosatisfy the design constraints. Alternatively, the function may be agenerating kernel that locally satisfies the desired property ofergodicity.

Once a generating function is identified, the function may be replicatedin two dimensions in order to fill the surface. Any 2D curved surfacecan be represented by a 2D array of values representing the height ofthe surface. For example, the pixel values in FIG. 22 represent theheight of the surface at each point. To fill the entire array, multiplecopies of the generating function may be tiled. Two straightforward andcomputationally inexpensive methods to fill such an array with multiplecopies of the generating function are substitution and addition.Substitution consists of replacing the pixel values within a section ofthe final array with the pixel values of the generating function. Inregions where two generating functions might overlap, one of them may betruncated. FIG. 24A is a graph 2400 illustrating truncated overlappingfunctions 2410 and 2420. Overlapping results in a vertical facet 2430that must be corrected to prevent sources of multiple scattering. Thiscan be accomplished by replacing the vertical facet with a sloped facet2440 that directs light out of the diffusion locus, i.e., with slopelarger than θ_(c) but still sufficiently small enough to prevent asecond reflection. Addition consists of adding the pixel values of thegenerating function to the pixel values of the total array. FIG. 24B isa graph 2450 illustrating addition of the pixel values of the generatingfunctions 2460 and 2470 to the pixel values of the total array. Note,that the height of the generating function is negative and that height=0is denoted by line 2490. To ensure smooth, continuous transition, theheight and slope of the generating function should approach zero at theboundary 2480. The advantage of this technique is that there are nofacets at the boundaries and so it is in principle possible to makebetter use of the available light. However, on average, addition leadsto a decrease in the average aspect ratio and thus an increase in thegain of the diffuser and thus must be corrected for as discussed below.

Informed by the exploration of random diffusers, even when thegenerating scattering feature has a cutoff angle that avoids a secondreflection with the surface, multiple reflections occur when two peaksinteract in a predictable way (i.e., when they exactly overlap or whenthey approach too closely). With Gaussian statistics, these situationsare likely to occur for some finite percentage of features. Therefore, astraightforward technique to increase the contrast is to modify thestatistics of the peak positions in an essentially non-Gaussian way toprevent the undesired events. The simplest way to do this is to limitthe random translation of the peaks so that they cannot overlap. Thetranslation must still be large enough to statistically hide theunderlying hexagonal character of the lattice.

The derivation of a 2D diffuser lens is slightly more complicated thanrevolving the 1D curve. Assuming an axially symmetric distribution, thedifferential equation describing θ as a function of r is:

$\begin{matrix}{\frac{flux}{steradian} = {\frac{I_{o}r{\mathbb{d}r}}{\sin\;\theta{\mathbb{d}\theta}} = {\alpha\;{D(\theta)}}}} & \left. 3 \right)\end{matrix}$where

(θ) is the desired distribution function, I_(o) is the incident flux perunit area, and α is a proportionality constant. To compute α, the totalflux incident on the generating kernel is equal to the integral of thedistribution function over all solid angles:

I_(o)π(r_(M)² − r_(m)²) = 2π∫_(o)^(θ_(c))α D(θ)sin  θ𝕕θwhere r_(m) is the inner radius of the generating kernel (which mayequal 0), r_(M) is the outer radius of the generating kernel, and θ_(c)is the cutoff angle of the distribution function. Once θ is known as afunction of r, the next step is to integrate the slope of the surface tofind the surface height:

$\begin{matrix}{\frac{\mathbb{d}z}{\mathbb{d}r} = {\tan\frac{\theta}{2}}} & \left. 5 \right)\end{matrix}$

In general, this integration is quite difficult to perform analyticallybut may be accomplished numerically relatively easily. FIGS. 25A to 25Care graphs of several exemplary solutions to equations 3 to 5 for

(θ)=cos(θ), i.e. a generating kernel or Lambertian diffuser. FIG. 25A isa graph 2500 of a solution generated in a circular region of maximumradius r_(M)=1, with the maximum slope on its outer boundary. FIG. 25Bis a graph 2510 of an exemplary solution generated in an annular regionwith inner radius, r_(m)=1 and outer radius r_(M)=1.5. The maximum slopein region 2510 occurs on the inner surface and so the two solutions canbe seamlessly joined together to form solution 2550. FIG. 25C is a graph2550 of an exemplary generating kernel solution generated by combiningsolutions 2500 and 2510. Following this procedure, arbitrary diffusionprofiles,

(θ), are possible, subject to the aforementioned limitations on cutoffangle.

FIG. 26 is a graph 2600 illustrating the gain simulated vianon-sequential ray tracing for a 2D Lambertian generating kernel.

FIG. 27 is a graph 2700 of the radially-averaged gain for the generatingkernel of FIG. 26. Completely filling a screen with a solution 2550 (asshown in FIG. 25C) presents the problems of substantially eliminatingempty space, but with minimal distortion to the generating kernel.

Tiling the Generating Kernels

One way to fill a screen with the engineered generating kernels is totile the generating kernels in a lattice configuration, e.g., a square,hexagonal, or any other regular shape lattice.

FIG. 28 is a schematic diagram illustrating an exemplary hexagonallattice 2800 configuration. As discussed above, however, empty space isundesirable in order to optimally use the available light and to preventan increase in the specular reflectivity (“spike” in the reflectivity).To substantially eliminate empty space, generating kernels 2802 in ahexagonal lattice 2800 may undergo overlapping. For example, toeliminate empty space on a screen using a hexagonal lattice 2800 ofgenerating kernels with a unit cell diameter of 2/√3, approximately20.9% of the unit cell areas would be overlapping.

FIG. 29 is a schematic diagram illustrating the unit cell overlap 2910of a hexagonal lattice 2900 of generating kernels.

FIG. 30 is a diagram illustrating the unit cell overlap 3010 of a squarelattice 3000. A square lattice 3000 may require an additional smallerunit cell 3020 to fill space, with the smaller unit cell's radius afunction of the large cell's 3030 radius. In the square lattice 3000configuration shown in FIG. 30, a 17.9% overlap is optimum. As discussedbelow, overlap of the generating kernels modifies the gain of thecombined structure. The change in gain is a function of the center tocenter distance of the nearest-neighbor individual generating kernelwhich, in turn, is a function of the azimuth within the lattice.Therefore, a perfect lattice has a deviation in the scattereddistribution reflecting the local arrangement of generating kernels. Forexample, a hexagonal lattice has six-fold symmetry in which the nearestneighbors of a given point are distributed around that point every 60°.Consequently, the scattered distribution will have an azimuthalmodulation with a periodicity of 60° whose amplitude is proportional tothe amount of overlap of the generating kernels. Regular lattices ofgenerating kernels may lead to moiré, diffraction, and other undesirableeffects. Modifying the regular lattice to achieve more randomization,for example, by using a hexagonal lattice with randomized latticepoints, reduces these effects. Additional overlap may result fromrandomizing a regular lattice. Also, the sizes of the unit cells may berandomized in addition to their positions. However, in this case itbecomes nearly impossible to pre-correct for overlap of the structures.

FIG. 31 is a schematic diagram of a hexagonal lattice 3100 allowing 0.1l of randomization of the center point. This configuration yields anoverlap area 3110 of 60%.

FIG. 32 is a diagram 3200 of a hexagonal lattice with smaller cells 3220dispersed in between the larger cells 3230, resulting in a much smalleroverlap area 3210 of 9.4%. The cell arrangement in FIG. 32 allows formore randomization.

Other techniques may also be used to decrease the effect of the lattice.For example, by using a larger unit cell comprised of multiplegenerating kernels for the initial tiling, the amount of randomizationnecessary to hide the lattice structure in the scattering profile may bedecreased. The hexagonal lattice has a six-fold rotational symmetry butif elements of two separate hexagonal lattices are combined with a 30°rotation between them, then the symmetry is increased to 12-fold. Thiscan be accomplished by any number of infinite sets of semi- anddemi-regular tessellations. Semi- and demi-regular tessellationstypically comprise multiple polygon (e.g., triangles and squares) andthus feature sizes to fill the lattice, providing additional variationof the height and orientation of the scattering features and reducesinterference. FIG. 33A is a schematic diagram 3300 illustrating asemi-regular tessellation pattern. FIG. 33B is a schematic diagram 3350illustrating the unit cell of this tiling, which consists of twohexagonal lattice components 3360 and 3370 as well as three squarelattices 3380. The vertices of the polygons indicate the centers ofgenerating functions. The angular orientation of nearest-neighbordirections in this tessellation is 0, 30, 60, 90, 120 . . . 330 incontrast to 0, 60, 120 . . . 300 for the regular hexagonal lattice.Furthermore, the square elements introduce an additional set of angles:15, 45, 75 . . . 345. FIGS. 33C and 33D are schematic diagramsillustrating vertices of the polygons 3390 and 3396 (and centers of thegenerating functions). The repeat distance of the structure is stillsignificantly smaller than a pixel.

Arbitrarily large pseudo-random tilings can be generated by performing a2D monte-carlo simulation of crystal melting. Such methods are wellknown for the study of interactions of hard disks as well as particleswith arbitrary attractive/repulsive interaction potentials. The startingpoint of the simulation is to generate a 2D lattice of particles on aregular grid. A random particle is then picked from the ensemble andtranslated by a small fixed amount. If the translation results in aregion that is not covered by any disk, then the move is rejected. Ifthe translation results in a decrease in the total amount of overlap ofthe particles then it is accepted. If the total amount of overlapincreases, then the move is accepted with a probability that isinversely proportional to the amount of increase. This process isrepeated until the system reaches equilibrium. Typically, when packingconstraints are emphasized in such a simulation, it will converge to ahexagonal lattice. Therefore, in order to suppress crystallinity whileminimizing the overlap of particles, it is useful to add a random changein the size of the particle to the monte-carlo step (subject to similarconstraints).

FIG. 34 is a schematic diagram 3400 of randomization via horizontaldisplacement. Uniformly generated displacements may ensure that thesurface is fully covered, as shown in 3400.

FIG. 35 is a graph 3500 of the probability distribution for cell centerto cell center displacement for a surface with randomized horizontaldisplacement as shown in FIG. 34. As shown in graph 3500, the maximumdisplacement for adjacent structures is 2.0 and the mean displacement is0.905.

A generic structure, for example, a Gaussian peak, does not necessarilyproduce the ideal scattering distribution. Instead, the statistics ofthe size and position of the structure may be controlled such that thedistribution is achieved by some relatively large number of features.For cases where it is difficult to specifically engineer the unit cellof a diffuser, for example, holographic or etched diffusers, a genericfeature such as a Gaussian peak is useful for determining designparameters. One cell, generating kernel, or microstructure comprising aplurality of generating kernels, would ideally have a slope approachingzero at a certain cut off radius, in order to allow cells to be stitchedtogether seamlessly. An ideal generating kernel, an engineeredgenerating kernel, or microstructure comprising a plurality ofgenerating kernels, would also preferably be ergodic, in that thegenerating kernel individually produces the entire desired distributionfunction. A generating kernel that produces a Lambertian distribution islocally ergodic. An ergodic generating function helps ensure thatintensity variations over the surface of the diffuser are minimized.

The diffusion features of an isolated Gaussian peak are modeled by thefollowing equation:z(r)=z ₀ e ^(−r) ¹The maximum slope of the isolated Gaussian peak occurs at r=½. Themaximum reflected angle is:θ_(max)=2 tan⁻¹(√{square root over (2)}z ₀ e ^(−1/2))To produce a cut-off slope of θ_(C), select z₀:

$z_{0} = {\frac{\tan\left( {\theta_{C}/2} \right)}{\sqrt{2}}\underset{/}{\left. {\mathbb{e}}^{{- 1}/2} \right)}}$

Again, adjacent peaks may cause double reflections depending on θ_(C)and proximity, but for θ_(C) less than ˜80°, there are no multiplereflections (except for peaks which overlap), as shown in FIG. 20.

FIG. 36 is a graph 3600 of diffusion angle as a function of separationof Gaussian peaks.

FIG. 37 is a graph 3700 illustrating the cutoff angle of a structurewith overlapping Gaussian features.

In the case of the engineered generating kernel, overlapping unit cellsand randomizing the location of unit cells may result in similarproblems disclosed above, for example, double reflections and changingof the gain profile. By choosing the additive method of placing unitcells within the array, the chance of double reflections issubstantially eliminated, leaving the change in gain profile to correct.

FIGS. 38A and 38B are graphs 3800 and 3850 illustrating engineeredLambertian diffuser overlap in two sample configurations. Lines 3810,3811, and 3812 represent Lambertian surfaces. Line 3820 represents thesum of the lines 3811 and 3812, indicating that the gain of thesum-surface is too high in configuration 3800. When the generatingkernels are moved even closer together in configuration 3850, line 3820is pushed even further down, indicating that the gain is even higher.

One way to address this problem is to pre-correct the generating kernelsfor overlap. FIGS. 39A and 39B illustrate one method of pre-correctingthe generating kernels to address overlap. Generating kernel A,generated by revolving line a, is ergodic; generating kernel B,generated by revolving line b, is ergodic; and generating kernel C,generated by revolving line c, is as close to ergodic as possible,subject to constraintc′(r _(max))=0whereb(r)+b(r−l)=c(r); (r<l/2).b′(r)+b′(r _(Max)−(r−r _(Min)))=c(r)(b′(r)−b(r))²+(b′(r _(Max)−(r−r _(Min)))−b(r _(Max) −r _(Min))))².

FIG. 40 is a graph 4000 of the pre-corrected cell with Lambertiandiffuser overlap. Lines 4010 and 4020 represent the target diffusershape. Line 4030 is the solution that produces line 4020 whileminimizing deviation from line 4010. Although the calculation did notcompletely pre-correct an arbitrary gain profile for overlap, theresults are highly sufficient. In this case, the slope is zero at theedges of the generating kernel. The gain profile has a sharp cutoff at80°, which results in substantially no diffusion beyond 80°.

To the extent that pre-correction of the generating kernel profile foroverlap is not sufficient, an additional step may be taken to arrive atthe desired diffusion profile. FIG. 41A is a graph 4100 of an examplegain profile 4110 showing separately the contributions from theoverlapping regions 4120 and non-overlapping regions 4130 after thestructure has been randomized. To the extent that the pre-correction ofthe overlap region is imperfect, the total gain does not track thetarget gain as shown by the lines 4110. The non-overlap regions may alsobe pre-corrected to account for the error. FIG. 41B shows a graph of anexample gain profile in which the overlapping and non-overlappingregions have complementary corrections such that the total gain is equalto the target gain. The total gain can be writtenG(θ)=G _(a)(θ)A _(o) +G _(b)(θ)A _(b),where G(θ) is the target gain, G_(a)(θ)/G_(b)(θ) are the gain curvesassociated with non-overlapping and overlapping regions of thegenerating kernel, and A_(a)/A_(b) are the areas of non-overlapping andoverlapping regions respectively. To the extent that G_(b)(θ)≠G(θ), wecan solve for a corrected gain contribution in the non-overlappingregion, G′_(a)(θ)

${G_{a}^{\prime}(\theta)} = {\frac{{G(\theta)} - {{G_{b}^{\prime}(\theta)}A_{b}}}{A_{a}}.}$Equations 3-5 must then be solved in order to find the correct shape forthe region of the generating kernel not subject to overlap. The sum 4160of both pre-corrected regions, the overlap 4170 and non-overlap 4180regions, substantially matches the target gain profile. In contrast tothe iterative design method described by Morris, this is a deterministicprocedure that reaches the optimum design in a fixed number of steps,i.e. design of generating kernel in overlap region, pre-correction ofgenerating kernel in overlap region, and design of generating kernel innon-overlap region.

The use of an engineered generating kernel as described abovesignificantly reduces the grainy appearance problem associated withconventional silver screens. Because each individual generating kernelmaps substantially the entire diffusion profile, large-scale spatialfluctuations due to statistical variation are largely avoided, even forlarge scattering angles, i.e., each individual generating kernel has atleast two regions contributing to the intensity at any given azimuthalangle.

Randomizing the height of the surface of the diffuser by adding someform of noise can be used to address the problem of coherent specklenear the retro-reflection direction. The amplitude of this randomizationshould be some small multiple of an optical wavelength in order toscramble the phase of the reflected light.

Practical benefits and considerations associated with providing a screenin accordance with the present disclosure include minimum cost per unitarea, spatial uniformity in performance, consistent performancereliability in manufacturing, and robustness in handling and cleaning.

The cost of screen material can be minimized by leveraging existingroll-to-roll processes as much as possible. The infrastructure tomanufacture roll-stock of optimized screen material may includeapparatus that performs micro-embossing, metallization, transparentdielectric (hard-coat) coatings, precision slitting and perforation (foracoustic transmission). There currently exist roll-to-roll embossingprocesses that are free of (transverse-direction) down-web seamsassociated with conventional nickel shims. According to the preferredmanufacturing process, UV embossing with a seamless embossing drumproduces continuous diffuser material. According to this process,diffuser roll stock is easily converted to finished screens by joiningsuch strips. Using precision roll-to-roll slitting, such strips can bejoined with butt-joints giving sufficiently small gaps that seams aresubstantially not visible in the theatre. With this approach, a back-endfilm joining process may be used to manufacture finished cinema screens.This process should preferably provide sufficient joint strength andreliability when the screen is mounted or stretched on the frame. Theseam (and any surrounded area impacted by the joining process) shouldpreferably be sufficiently small that it is not observable to theaudience.

A potential benefit of film joining prior to coating (e.g., spraypainting metal flake) is that optically thick layers can planarize smallfeatures. In practice, joins in such screens are frequently observablebecause there is an asymmetric “step” in the seam. Because of the highspecular reflection, the associated macroscopic facet creates a largedisruption in the angular dispersion of light. When the material is flatacross the join (e.g., like a butt-joint), seams are generally notobservable when the gap is below approximately 50 microns, and in mostinstances, up to 100 microns. This can probably be even larger whenfurther steps are done to mask the join, such as randomizing the edgeprofile.

In the event that precision roll-to-roll slitting does not providesufficient accuracy, another preferred technique for manufacturing thefinished screen from coated strips, where strips hang vertically forstrength reasons, is to simultaneously slit the sheets so that they canbe easily butted together. This can be done by overlapping the sheetsand using a single knife, or using a pair of knives with fixedseparation. While this gives relief from edge straightness constraints,it is a batch process that is substantially more labor intensive thanthe precision roll-to-roll slitting approach.

After slitting, the two sheets can be butted together, either by drivingthe sheets together locally with a system of rollers of suitableprofile, or by translating a sheet using a vacuum table to globally buttthe materials together. With the coated surfaces face down, the filmscan be joined together using any one of several methods, includingadhesives, chemicals, or welding processes. Adhesives can include UVcures, e-beam cure, or various thermo-set processes. Chemical bondingcan include solvents, or doped solvents. Welding processes can includevarious means of delivering thermal energy to the join, preferablylasers.

Given the lack of surface area associated with a butt-joint, it islikely that additional mechanical support may be used to ensure jointstrength. This can be provided using some form of backer-strip, whichcreates a T-joint. The thickness and size of the backer strip can beselected to ensure that the front surface of the finished (stretched)screen is uniform across the boundary. In some instances, it may bepreferable to laminate the entire screen to a secondary backer sheet,such as a fabric, which further improves the strength and appearance.

A more sophisticated screen design according to the present disclosuremay involve local (position specific) control of diffusion properties.This can be done by manufacturing rolls of material that are dedicatedto specific locations on the screen. Typically, this involves a biasangle in the diffusion direction, or in the case of a Lambertian-likescreen, a (first-order) bias in the location of the centroid of thelocus.

Assuming that the screen stock is made using roll-to-roll processing,and that the strips are again hung vertically, the local correction islikely to be in the horizontal direction. The design of the cross-webdiffusion profile of any sheet can vary adiabatically, such that thereis no abrupt change in diffusion profile at the boundary between sheets.This allows very large screens with quasi-continuously varying optimizeddiffusion properties in the horizontal direction.

A screen manufactured as described above could have the same performanceas a screen curved about the vertical, but in a flat format. Moreover,the effective performance of a compound-curved (e.g., toroidal) screencould be achieved by curving the aforementioned screen about thehorizontal axis. This eliminates the complications of manufacturing alarge compound curved screen (e.g., pulling vacuum on the volume behindflat/flexible screen material).

While various embodiments in accordance with the principles disclosedherein have been described above, it should be understood that they havebeen presented by way of example only, and not limitation. Thus, thebreadth and scope of the invention(s) should not be limited by any ofthe above-described exemplary embodiments, but should be defined only inaccordance with any claims and their equivalents issuing from thisdisclosure. Furthermore, the above advantages and features are providedin described embodiments, but shall not limit the application of suchissued claims to processes and structures accomplishing any or all ofthe above advantages.

Additionally, the section headings herein are provided for consistencywith the suggestions under 37 CFR 1.77 or otherwise to provideorganizational cues. These headings shall not limit or characterize theinvention(s) set out in any claims that may issue from this disclosure.Specifically and by way of example, although the headings refer to a“Technical Field,” the claims should not be limited by the languagechosen under this heading to describe the so-called field. Further, adescription of a technology in the “Background” is not to be construedas an admission that certain technology is prior art to any invention(s)in this disclosure. Neither is the “Summary” to be considered as acharacterization of the invention(s) set forth in issued claims.Furthermore, any reference in this disclosure to “invention” in thesingular should not be used to argue that there is only a single pointof novelty in this disclosure. Multiple inventions may be set forthaccording to the limitations of the multiple claims issuing from thisdisclosure, and such claims accordingly define the invention(s), andtheir equivalents, that are protected thereby. In all instances, thescope of such claims shall be considered on their own merits in light ofthis disclosure, but should not be constrained by the headings set forthherein.

What is claimed is:
 1. A projection screen, comprising: a contouredreflective surface having a predetermined scatter profile, thepredetermined scatter profile being operable to reflect incident lightfrom a predetermined incident angle range to within a diffusion locus,wherein the light reflected from the predetermined incident angle rangeto within the diffusion locus substantially undergoes no more than onereflection from the contoured reflective surface, wherein the diffusionlocus is defined by a region in which reflections within the diffusionlocus substantially undergo no more than one reflection on the contouredreflective surface, and wherein light reflected outside of the diffusionlocus undergoes one or more reflections on the contoured reflectivesurface.
 2. The projection screen of claim 1, wherein the incident lightis polarized light, and wherein light reflected from the predeterminedincident angle range to within the diffusion locus maintains the samestate of polarization.
 3. The projection screen of claim 1, wherein thecontoured reflective surface comprises a plurality of generatingkernels.
 4. The projection screen of claim 3, wherein each generatingkernel substantially satisfies the predetermined scatter profile.
 5. Theprojection screen of claim 4, wherein each generating kernel is operableto minimize point-to-point variation in diffusion.
 6. The projectionscreen of claim 3, wherein the plurality of generating kernels aredistributed to optimize viewing from a location in the diffusion locus.7. The projection screen of claim 3, wherein the plurality of generatingkernels satisfy the predetermined scatter profile.
 8. The projectionscreen of claim 7, wherein the plurality of generating kernels isoperable to minimize point-to-point variation in diffusion.
 9. Theprojection screen of claim 3, wherein the plurality of generatingkernels satisfies a statistical model to minimize double reflectionsbetween generating kernels.
 10. The projection screen of claim 3,further comprising a dielectric overcoat, the dielectric overcoatdistributed over the plurality of generating kernels.
 11. The projectionscreen of claim 3, wherein the plurality of generating kernels aredistributed in a substantially regular lattice.
 12. The projectionscreen of claim 11, wherein the substantially regular lattice comprisesa hexagonal lattice.
 13. The projection screen of claim 11, wherein theplurality of generating kernels are distributed in a tessellationpattern.
 14. The projection screen of claim 11, wherein thesubstantially regular lattice comprises randomized centers.
 15. Theprojection screen of claim 11, wherein at least two of the generatingkernels are substantially overlapping.
 16. The projection screen ofclaim 11, wherein at least one of the generating kernels is disposed topre-correct for an offset resulting from overlapping adjacent generatingkernels.
 17. The projection screen of claim 1, wherein the diffusionlocus is defined by a predetermined reflection angle range.
 18. Theprojection screen of claim 1, wherein substantially all light undergoingmore than one reflection is distributed outside of the diffusion locus.19. The projection screen of claim 1, wherein the diffusion locusincludes substantially all viewing locations in an auditorium.
 20. Theprojection screen of claim 1, wherein light reflected to the diffusionlocus is of enhanced brightness.
 21. The projection screen of claim 1,wherein light reflected to the diffusion locus is substantially uniform.22. The projection screen of claim 1, wherein light reflected to thediffusion locus is of enhanced contrast.
 23. The projection screen ofclaim 1, wherein light reflected to the diffusion locus satisfies apredetermined gain profile.
 24. A method for providing a projectionscreen, the method comprising: providing a contoured reflective surfacehaving a predetermined scatter profile, the predetermined scatterprofile being operable to reflect incident light from a predeterminedincident angle range to within a diffusion locus, wherein the lightreflected from the predetermined incident angle range to within thediffusion locus substantially undergoes no more than one reflection fromthe contoured reflective surface, further comprising defining thediffusion locus by a region in which reflections within the diffusionlocus substantially undergo no more than one reflection on the contouredreflective surface, and wherein light reflected outside of the diffusionlocus undergoes one or more reflections on the contoured reflectivesurface.
 25. The method of claim 24, wherein the incident light ispolarized light, and wherein light reflected from the predeterminedincident angle range to within the diffusion locus maintains the samestate of polarization.
 26. The method of claim 24, wherein the contouredreflective surface comprises a plurality of generating kernels.
 27. Themethod of claim 26, further comprising satisfying the predeterminedscatter profile with each generating kernel.
 28. The method of claim 27,further comprising minimizing point-to-point variation in diffusion witheach generating kernel.
 29. The method of claim 26, further comprisingsatisfying the predetermined scatter profile with the plurality ofgenerating kernels.
 30. The method of claim 29, further comprisingminimizing point-to-point variation in diffusion with the plurality ofgenerating kernels.
 31. The method of claim 26, further comprisingsatisfying a statistical model to minimize double reflections betweengenerating kernels with the plurality of generating kernels.
 32. Themethod of claim 26, further comprising distributing the plurality ofgenerating kernels to optimize viewing from a location in the diffusionlocus.
 33. The method of claim 26, further comprising: providing adielectric overcoat; and distributing the dielectric overcoat over theplurality of generating kernels.
 34. The method of claim 26, furthercomprising distributing the plurality of generating kernels in asubstantially regular lattice.
 35. The method of claim 34, furthercomprising randomizing the centers of the substantially regular lattice.36. The method of claim 34, further comprising overlapping at least twoof the generating kernels.
 37. The method of claim 26, furthercomprising distributing the plurality of generating kernels in asubstantially hexagonal lattice.
 38. The method of claim 26, furthercomprising distributing the plurality of generating kernels in atessellation pattern.
 39. The method of claim 26, further comprisingdisposing at least one of the generating kernels to pre-correct for anoffset resulting from overlapping adjacent generating kernels.
 40. Themethod of claim 24, further comprising defining the diffusion locus by apredetermined reflection angle range.
 41. The method of claim 24,further comprising eliminating substantially all light undergoingmultiple reflections from being distributed within the diffusion locus.42. The method of claim 24, further comprising distributingsubstantially all light undergoing more than one reflection outside ofthe diffusion locus.
 43. The method of claim 26, further comprisingincluding substantially all viewing locations in an auditorium in thediffusion locus.
 44. The method of claim 24, further comprisingenhancing the brightness of the light reflected to the diffusion locus.45. The method of claim 24, further comprising making substantiallyuniform the light reflected to the diffusion locus.
 46. The method ofclaim 24, further comprising enhancing the contrast of the lightreflected to the diffusion locus.
 47. The method of claim 24, furthercomprising satisfying a predetermined gain profile with the lightreflected to the diffusion locus.
 48. A projection screen, comprising: acontoured reflective surface having a predetermined scatter profile, thepredetermined scatter profile being operable to reflect incident lightfrom a predetermined incident angle range to within a diffusion locus,wherein the light reflected from the predetermined incident angle rangeto within the diffusion locus substantially undergoes no more than onereflection from the contoured reflective surface, and wherein thecontoured reflective surface is operable to substantially eliminatelight undergoing multiple reflections from being distributed within thediffusion locus.